课程： 4.15有理数的比较

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选择节有理数的比较

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有理数的比较
Christine is working on a craft project and she needs ribbon. She heads to the fabric store where they are having a sale on ribbon remnants (leftover scraps of ribbon). Everything in the ribbon remnants bin is only $2. She wants to buy one ribbon remnant but wants to get the longest piece of ribbon she can. She sees some ribbon is $\begin{aligned} \frac{2}{3} \end{aligned}$ of a yard, some ribbon is $\begin{aligned} \frac{7}{8} \end{aligned}$ of a yard, and some ribbon is 0.8 of a yard. Which ribbon should Christine buy if she wants the most ribbon?
::克里斯蒂娜正在做一个工艺品项目,她需要丝带。她正前往布料店销售丝带残留物(剩余丝带)。丝带残留物中的所有物品只有2美元。她想买一个丝带残留物,但想得到她能买到的最长的丝带。她看到一些丝带是院子的23个,有些丝带是院子的78个,有些丝带是院子的0.8个。如果Christine想要最好的丝带,应该买哪条丝带?

In this concept, you will learn how to order on a number line and how to using inequality symbols.
::在此概念中,您将学会如何按数字线排序,以及如何使用不平等符号。

Comparing Rational Numbers
::比较理性数字

The rational numbers are the set of numbers that can be written as a ratio of two integers. Integers, fractions, repeating decimals, and terminating decimals are all rational numbers.
::理性数字是可以写成两个整数之比的一组数字。整数、分数、重复的十进制数和终止的十进制数都是合理的数字。

Sometimes you will want to compare different rational numbers or put them in order from smallest to largest. A number line can help you to do this. On a number line, the further a number is to the right, the larger its value. The further a number is to the left, the smaller its value.
::有时,您会想要比较不同的理性数字,或者排列它们从最小到最大。数字行可以帮助您做到这一点。在数字行中,数字数越往右,其值就越大。数字越往左,其值就越小。

Once you know how rational numbers are related, you can use the following inequality symbols to describe their relationship:
::一旦知道合理数字是如何关联的,您就可以使用以下不平等符号来描述他们之间的关系:

> means is greater than .
::> 手段大于。

< means is less than .
::< 手段小于。

= means is equal to .
::手段等于。

Let's look at an example.
::让我们举个例子。

Choose the inequality symbol that goes in the blank to make this statement true.
::选择空白中的不平等符号以使此语句变为真实。

$\begin{aligned} - 2.5 \underline{} - 5 \end{aligned}$

First, draw a number line from -5 to 5. Place the numbers -2.5 and -5 on that number line. -2.5 will be between -2 and -3 on the number line.
::首先,从 -5 到 5 绘制一个数字行。将数字 - 2.5 和 - 5 放在该数字行上。 - 2.5 将在 - 2 和 - 3 在数字行上。

Since -2.5 is further to the right on the number line compared to -5, -2.5 is greater than -5.
::因为-2.5是在数字线右侧,而5-5-2.5大于-5。

The answer is $\begin{aligned} - 2.5 > - 5 \end{aligned}$ .
::答案是-2.5 >-5。

Let's look at another example.
::让我们再举一个例子。

Order these rational numbers from least to greatest.
::把这些理性数字从最小到最大顺序排列。

$\begin{aligned} \frac{4}{5} 0.6 1 0. \bar{6} \end{aligned}$

When comparing rational numbers it can help to have all your numbers in the same form. It is usually easiest to compare numbers when they are in decimal form. Convert $\begin{aligned} \frac{4}{5} \end{aligned}$ to its decimal form.
::当比较理性数字时, 它可以帮助将您的所有数字都以同样的形式出现。当数字以小数形式出现时, 它通常比较容易。将 4 5 转换为小数形式。

$\begin{aligned} \frac{4}{5} = 0.8 \end{aligned}$

Next, plot all four numbers on a number line. When dealing with decimals it can help to draw a number line that is divided into tenths. So, draw a number line from 0 to 1 and divide it into tenths.
::接下来,在数字行上绘制所有四个数字。当处理小数点时,它可以帮助绘制一个数字行,该数字行分为十分。因此,将数字行从0到1绘制成数字行,然后将其分为十分。

Notice that $\begin{aligned} 0. \bar{6} \end{aligned}$ appears between the 0.6 and the 0.7 on the number line.
::注意数字线0.6和0.7之间出现0.6和0.7之间

Now, with the four numbers plotted on the number line you can see that the smallest number is 0.6 because it is furthest to the left. The largest number is 1 because it is furthest to the right.
::现在,在数字线上绘制了四个数字,你可以看到最小的数字是 0.6,因为它离左最远。最大的数字是 1,因为它离右最远。

$\begin{aligned} 0.6 < 0. \bar{6} < \frac{4}{5} < 1 \end{aligned}$

The answer is that ordered from least to greatest, the numbers are $\begin{aligned} 0.6, 0. \bar{6}, \frac{4}{5}, 1 \end{aligned}$ .
::答案是从最少到最多订购, 数字是0.6, 0. 6', 4 5, 1'.

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Christine, who was at the fabric store.
::早些时候,有人给了你一个问题关于Christine, 谁在布料店。

She is trying to figure out which ribbon to buy. She wants the most ribbon for her money. Her three options are:
::她想要最美的丝带

$\begin{aligned} \frac{2}{3} \end{aligned}$ of a yard of ribbon for $2
::2块2个3个丝带院子 $2

$\begin{aligned} \frac{7}{8} \end{aligned}$ of a yard of ribbon for $2
::7个8个彩带院子 $2 $2

0.8 of a yard of ribbon for $2
::0.8块彩带院子 $2

Christine needs to decide which piece of ribbon is the longest. She is trying to compare the rational numbers $\begin{aligned} \frac{2}{3}, \frac{7}{8} \end{aligned}$ and 0.8 to figure out which number is the biggest.
::克里斯蒂娜需要决定哪个丝带最长。她试图比较理性数字23、7、8和0.8, 以找出哪一个是最大的。

To help Christine, you should first make all numbers be in the same form. It is often easiest to compare numbers in decimal form, so you should convert the fractions to decimals. You can also write the decimals so they each have the same number of digits after the decimal point to make the comparison easier.
::要帮助Christine, 您首先应该将所有数字都以同样的形式排列。通常比较小数形式的数字比较容易, 所以您应该将分数转换为小数。您也可以写小数, 这样小数点之后, 每一个小数点都有相同的数字, 以便于比较。

$\begin{aligned} \frac{2}{3} = 0.66 \bar{6} \end{aligned}$

$\begin{aligned} \frac{7}{8} = 0.875 \end{aligned}$

$\begin{aligned} 0.8 = 0.800 \end{aligned}$

Looking at it this way, you can see that 0.875 is the largest number and $\begin{aligned} 0.66 \bar{6} \end{aligned}$ is the smallest number. If Christine wants to get the most amount of ribbon, she should buy the ribbon that is $\begin{aligned} \frac{7}{8} \end{aligned}$ of a yard.
::这样看,你可以看到0.875是最大的数字,0.66 6'是最小的数字。如果克莉丝汀想得到最多数量的丝带,她应该买到一个院子的78个丝带。

The answer is that Christine should buy the $\begin{aligned} \frac{7}{8} \end{aligned}$ of a yard of ribbon if she wants the most ribbon.
::答案是如果Christine想要最好的丝带她应该买下78个丝带院子

Example 2
::例2

Compare the following rational numbers.
::比较下列合理数字。

$\begin{aligned} - \frac{4}{5} \end{aligned}$ and -0.25
::- 4 5和-0.25

First, put both rational numbers into the same form. Let's convert $\begin{aligned} - \frac{4}{5} \end{aligned}$ to a decimal. Divide 4 by 5 to find that $\begin{aligned} \frac{4}{5} = 0.8 \end{aligned}$ . So you have
::首先,将两个理性数字都以相同形式排列。让我们将 4 5 转换为小数点后一位数。除以 4 5 除以 5 以找到 4 5 = 0.8。因此,您可以将 4 5 转换为小数点后一位数。

$\begin{aligned} - \frac{4}{5} = - 0.8 \end{aligned}$

Next, plot both numbers on a number line. Draw a number line from -1 to 0 and divide it into tenths.
::下一步, 在数字行上绘制两个数字。从 - 1 到 0 绘制一个数字行, 并将其分成十分之一。

Notice that -0.25 appears between the -0.2 and the -0.3 on the number line.
::注意数字行的-0.2和-0.3之间显示 -0.25。

Now, with the two numbers plotted on the number line you can see that $\begin{aligned} - \frac{4}{5} \end{aligned}$ is the smaller number because is further to the left. The larger number is -0.25.
::现在,在数字线上绘制了两个数字,你可以看到 - 4 5是较小的数字,因为离左边更远。更大的数字是 - 0. 25。

The answer is that $\begin{aligned} - \frac{4}{5} < - 0.25 \end{aligned}$ .
::答案是-4 5 < -0.25。

Example 3
::例3

Compare the following rational numbers.
::比较下列合理数字。

-0.7 and $\begin{aligned} - \frac{7}{10} \end{aligned}$
::-0.7和--7 10

First, put both rational numbers into the same form. Let's convert $\begin{aligned} - \frac{7}{10} \end{aligned}$ to a decimal. Divide 7 by 10 to find that $\begin{aligned} \frac{7}{10} = 0.7 \end{aligned}$ . So you have
::首先,将两个理性数字都以相同形式排列。让我们转换为 -7 10 至小数点后一位。除以 7 乘 10 以找到 7 10 = 0. 7。因此,您有

$\begin{aligned} - \frac{7}{10} = - 0.7 \end{aligned}$

Next, you would usually draw a number line to help you to compare the numbers, but you can already see that the two numbers are equal! So drawing the number line is not necessary this time.
::下一步, 您通常会绘制一个数字行来帮助您比较数字, 但您已经可以看到这两个数字是相等的。因此, 此时不需要绘制数字行。

The answer is that $\begin{aligned} - 0.7 = - \frac{7}{10} \end{aligned}$ .
::答案是- 0. 7 =- 7 10。

Example 4
::例4

Compare the following rational numbers.
::比较下列合理数字。

0.34 and $\begin{aligned} \frac{1}{2} \end{aligned}$
::0.34和1 2

First, put both rational numbers into the same form. Let's convert $\begin{aligned} \frac{1}{2} \end{aligned}$ to a decimal.
::首先,将两个理性数字都放入相同的形式。让我们将12转换为小数点后一位。

$\begin{aligned} \frac{1}{2} = 0.5 \end{aligned}$

Next, plot both numbers on a number line. Draw a number line from 0 to 1 and divide it into tenths.
::下一步, 在数字行上绘制两个数字。从 0 到 1 绘制一个数字行, 并将其分成十分之一。

Notice that 0.34 appears between the 0.3 and the 0.4 on the number line.
::注意数字线0.3和0.4之间出现0.34。

Now, with the two numbers plotted on the number line you can see that 0.34 is the smaller number because is further to the left. The larger number is $\begin{aligned} \frac{1}{2} \end{aligned}$ .
::现在,在数字线上绘制了两个数字,你可以看到0.34是较小的数字,因为离左边更远。最大的数字是12。

The answer is that $\begin{aligned} 0.34 < \frac{1}{2} \end{aligned}$ .
::答案是0.34 < 1 2。

Example 5
::例5

Compare the following rational numbers.
::比较下列合理数字。

$\begin{aligned} 7.77 \bar{7} \end{aligned}$ and $\begin{aligned} 7 \frac{7}{10} \end{aligned}$
::7.77 7 7 和7 7 10

First, put both rational numbers into the same form. Let's convert $\begin{aligned} 7 \frac{7}{10} \end{aligned}$ to a decimal. You know that $\begin{aligned} \frac{7}{10} = 0.7 \end{aligned}$ , so you have
::首先,将两个理性数字都以相同形式排列。让我们将7 7 10 转换为十进制。你知道, 7 10 = 0. 7, 所以,

$\begin{aligned} 7 \frac{7}{10} = 7.7 \end{aligned}$ .

Next you could plot both numbers on a number line, but this time that might not be as easy or as helpful. Sometimes when comparing numbers in decimal form it can be helpful for each number to have the same number of digits after the decimal point.
::接下来,您可以在数字行中绘制两个数字,但这一次可能不那么容易或有用。有时,当以小数数形式比较数字时,每个数字在小数点之后具有相同的数字数是有用的。

You can rewrite 7.7 by adding zeros at the end without changing the value of the number.
::您可以在不改变数字值的情况下通过在结尾处添加零来重写 7.7 。

$\begin{aligned} 7.7 = 7.70 \bar{0} \end{aligned}$

Now you want to compare $\begin{aligned} 7.70 \bar{0} \end{aligned}$ with $\begin{aligned} 7.77 \bar{7} \end{aligned}$ . Focus on the digits after the decimal point. You know that 777 is greater than 700. This means that $\begin{aligned} 7.77 \bar{7} \end{aligned}$ is greater than $\begin{aligned} 7.70 \bar{0} \end{aligned}$ .
::注意小数点后的数字,你知道777大于700。这意味着7.777大于7.70。

The answer is that $\begin{aligned} 7.77 \bar{7} > 7 \frac{7}{10} \end{aligned}$ .
::答案是7.777 7 > 7 7 10

Review
::回顾

Choose the inequality symbol (>, <, or =) that goes in the blank to make each statement true.
::选择空白中的不平等符号 (>, < 或 =) ,以使每个语句都真实。

$\begin{aligned} 1 \underline{} 1 \frac{1}{10} \end{aligned}$

$\begin{aligned} - 2 \underline{} 1 \frac{1}{3} \end{aligned}$

$\begin{aligned} \frac{2}{5} \underline{} 0.3 \end{aligned}$

$\begin{aligned} - .34 \underline{} \frac{- 3}{10} \end{aligned}$

$\begin{aligned} 5.6 \underline{} 5.7 \end{aligned}$

$\begin{aligned} - 8 \underline{} - 12 \end{aligned}$

$\begin{aligned} - \frac{3}{4} \underline{} - .75 \end{aligned}$

$\begin{aligned} 18.4 \underline{} 18.2 \end{aligned}$

$\begin{aligned} - .356 \underline{} - 1 \frac{1}{10} \end{aligned}$

$\begin{aligned} 5.678888 \underline{} 6 \frac{5}{10} \end{aligned}$

$\begin{aligned} - .509 \underline{} \frac{- 509}{1000} \end{aligned}$

$\begin{aligned} .87 \underline{} \frac{8}{10} \end{aligned}$

$\begin{aligned} - 4 \underline{} - 14 \end{aligned}$

$\begin{aligned} 1.212 \underline{} 1.232 \end{aligned}$

Place each rational number on the number line below. Then list these rational numbers in order from least to greatest.
::将每个理性数字放在下面的数行中。然后列出这些理性数字, 从最小到最大顺序。

$\begin{aligned} \frac{1}{2} 0.9 0 0. \bar{9} \end{aligned}$

Review (Answers)
::回顾(答复)

章节大纲

常规

有理数的比较

Comparing Rational Numbers ::比较理性数字

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Comparing Rational Numbers
::比较理性数字

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)